Applications of local linking to nonlocal Neumann problems

2014 ◽  
Vol 17 (01) ◽  
pp. 1450001 ◽  
Author(s):  
Giovanni Molica Bisci ◽  
Vicenţiu D. Rădulescu
Keyword(s):  
Differential Operator ◽  
Neumann Problem ◽  
Weak Solutions ◽  
Linking Theorem ◽  
Multiplicity Result ◽  
Elliptic Differential Operator ◽  
Abstract Result ◽  
Existence Of Weak Solutions ◽  
Neumann Problems ◽  
Superlinear Growth

We study a nonlocal Neumann problem driven by a nonhomogeneous elliptic differential operator. The reaction term is a nonlinearity function that exhibits p-superlinear growth but need not satisfy the Ambrosetti–Rabinowitz condition. By using an abstract linking theorem for smooth functionals, we prove a multiplicity result on the existence of weak solutions for such problems. An explicit example illustrates the main abstract result of this paper.

scholarly journals Poincaré Inequalities and Neumann Problems for the p-Laplacian

2018 ◽  
Vol 61 (4) ◽  
pp. 738-753 ◽  
Author(s):  
David Cruz-Uribe ◽  
Scott Rodney ◽  
Emily Rosta
Keyword(s):  
Quadratic Form ◽  
Neumann Problem ◽  
Sobolev Spaces ◽  
Weak Solutions ◽  
Poincaré Inequalities ◽  
Existence Of Weak Solutions ◽  
Neumann Problems ◽  
Poincare Inequalities ◽  
Associated Matrix ◽  
Weighted Poincaré Inequalities

AbstractWe prove an equivalence between weighted Poincaré inequalities and the existence of weak solutions to a Neumann problem related to a degenerate p-Laplacian. The Poincaré inequalities are formulated in the context of degenerate Sobolev spaces defined in terms of a quadratic form, and the associated matrix is the source of the degeneracy in the p-Laplacian.

scholarly journals Duality by reproducing kernels

2003 ◽  
Vol 2003 (6) ◽  
pp. 327-395 ◽  
Author(s):  
A. Shlapunov ◽  
N. Tarkhanov
Keyword(s):  
Differential Operator ◽  
Neumann Problem ◽  
Compact Manifold ◽  
Reproducing Kernels ◽  
Elliptic Differential Operator ◽  
Regularity Theorem ◽  
Neumann Problems ◽  
Smooth Compact Manifold ◽  
Convexity Properties ◽  
The Neumann Problem

LetAbe a determined or overdetermined elliptic differential operator on a smooth compact manifoldX. Write𝒮A(𝒟)for the space of solutions of the systemAu=0in a domain𝒟⋐X. Using reproducing kernels related to various Hilbert structures on subspaces of𝒮A(𝒟), we show explicit identifications of the dual spaces. To prove the regularity of reproducing kernels up to the boundary of𝒟, we specify them as resolution operators of abstract Neumann problems. The matter thus reduces to a regularity theorem for the Neumann problem, a well-known example being the∂¯-Neumann problem. The duality itself takes place only for those domains𝒟which possess certain convexity properties with respect toA.

THREE NON-ZERO SOLUTIONS FOR ELLIPTIC NEUMANN PROBLEMS

2011 ◽  
Vol 09 (04) ◽  
pp. 383-394 ◽  
Author(s):  
GIUSEPPINA D'AGUÌ ◽  
GIOVANNI MOLICA BISCI
Keyword(s):  
Critical Point ◽  
Neumann Problem ◽  
Weak Solutions ◽  
Open Interval ◽  
Multiplicity Result ◽  
Neumann Problems ◽  
Critical Point Result

In this note we obtain a multiplicity result for an eigenvalue Neumann problem. Precisely, a recent critical point result for differentiable functionals is exploited, in order to prove the existence of a determined open interval of positive eigenvalues for which the problem admits at least three non-zero weak solutions.

scholarly journals Weak Solutions and Energy Estimates for a Class of Nonlinear Elliptic Neumann Problems

2013 ◽  
Vol 13 (2) ◽  
Author(s):  
Gabriele Bonanno ◽  
Giovanni Molica Bisci ◽  
Vicenţiu D. Rădulescu
Keyword(s):  
Neumann Problem ◽  
Laplace Operator ◽  
Weak Solutions ◽  
Existence Results ◽  
Energy Estimates ◽  
Lack Of Compactness ◽  
Neumann Problems ◽  
Estimates Of Solutions ◽  
Nonlinear Elliptic

AbstractWe establish existence results and energy estimates of solutions for a homogeneous Neumann problem involving the p-Laplace operator. The case of large dimensions, corresponding to the lack of compactness of W

Infinitely many solutions for non-homogeneous Neumann problems in Orlicz-Sobolev spaces

2018 ◽  
Vol 68 (4) ◽  
pp. 867-880
Author(s):  
Saeid Shokooh ◽  
Ghasem A. Afrouzi ◽  
John R. Graef
Keyword(s):  
Variational Methods ◽  
Critical Point ◽  
Neumann Problem ◽  
Sobolev Spaces ◽  
Weak Solutions ◽  
Critical Point Theory ◽  
Point Theory ◽  
Infinitely Many Solutions ◽  
Neumann Problems

Abstract By using variational methods and critical point theory in an appropriate Orlicz-Sobolev setting, the authors establish the existence of infinitely many non-negative weak solutions to a non-homogeneous Neumann problem. They also provide some particular cases and an example to illustrate the main results in this paper.

scholarly journals Eigenvalues for a Neumann Boundary Problem Involving thep(x)-Laplacian

2015 ◽  
Vol 2015 ◽  
pp. 1-5
Author(s):  
Qing Miao
Keyword(s):  
Variational Principle ◽  
Boundary Problem ◽  
Neumann Problem ◽  
Weak Solutions ◽  
Neumann Boundary ◽  
Ekeland’S Variational Principle ◽  
Laplacian Operator ◽  
Ekeland's Variational Principle ◽  
Existence Of Weak Solutions

We study the existence of weak solutions to the following Neumann problem involving thep(x)-Laplacian operator:  -Δp(x)u+e(x)|u|p(x)-2u=λa(x)f(u),in  Ω,∂u/∂ν=0,on  ∂Ω. Under some appropriate conditions on the functionsp,  e,  a, and  f, we prove that there existsλ¯>0such that anyλ∈(0,λ¯)is an eigenvalue of the above problem. Our analysis mainly relies on variational arguments based on Ekeland’s variational principle.

scholarly journals Existence of weak solutions to a certain homogeneous parabolic Neumann problem involving variable exponents and cross-diffusion

2020 ◽  
Vol 6 (2) ◽  
pp. 685-709
Author(s):  
Gurusamy Arumugam ◽  
André H. Erhardt
Keyword(s):  
Neumann Problem ◽  
Weak Solutions ◽  
Type Inequality ◽  
Neumann Boundary ◽  
Stability Estimate ◽  
Energy Estimates ◽  
Variable Exponents ◽  
Existence Of Weak Solutions ◽  
Cross Diffusion ◽  
Time Variables

Abstract This paper deals with a homogeneous Neumann problem of a nonlinear diffusion system involving variable exponents dependent on spatial and time variables and cross-diffusion terms. We prove the existence of weak solutions using Galerkin’s approximation and we derive suitable energy estimates. To this end, we establish the needed Poincaré type inequality for variable exponents related to the Neumann boundary problem. Furthermore, we show that the investigated problem possesses a unique weak solution and satisfies a stability estimate, provided some additional assumptions are fulfilled. In addition, we show under which conditions the solution is nonnegative.

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